The Topological Entropy of Powers on Lie Groups
Mauro Patr\~ao

TL;DR
This paper computes the topological entropy of power maps on Lie groups, extending known results from commutative cases to non-commutative groups using Lie theory, revealing that such entropy is positive for compact groups with discrete center.
Contribution
It generalizes the formula for topological entropy of power maps from commutative Lie groups to all Lie groups, including non-commutative cases, using structure theory.
Findings
Topological entropy of powers on Lie groups equals the dimension of a maximal torus times log(k).
For compact Lie groups with discrete center, the entropy is always positive.
Non-commutative cases differ from endomorphisms, which have zero entropy.
Abstract
This article addresses the problem of computing the topological entropy of an application , where is a Lie group, given by some power , with a positive integer. When is commutative, is an endomorphism and its topological entropy is given by , where is the maximal torus of , as shown in \cite{patrao:endomorfismos}. But when is not commutative, is no longer an endomorphism and these previous results cannot be used. Still, has some interesting symmetries, for example, it commutes with the conjugations of . In this paper, the structure theory of Lie groups is used to show that , where is a maximal torus of , generalizing the commutative case formula. In particular, the topological entropy of powers on compact Lie groups with discrete center is…
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Taxonomy
TopicsHomotopy and Cohomology in Algebraic Topology · Geometric Analysis and Curvature Flows · Geometry and complex manifolds
